Which of the following represents the solution to the inequality [tex]$2|5-2 x|-3 \leq 15$[/tex]?
A. [tex](-\infty,-2) \cup(7, \infty)$[/tex]
B. [tex](-\infty, 15) \cup(75, \infty)$[/tex]
C. [tex]$[-2,7]$[/tex]
D. [tex]$[1.5, 7.5]$[/tex]
Answer (2)
Isolate the absolute value term: 2∣5 − 2 x ∣ ≤ 18 becomes ∣5 − 2 x ∣ ≤ 9 .
Rewrite as a compound inequality: − 9 ≤ 5 − 2 x ≤ 9 .
Solve for x : − 14 ≤ − 2 x ≤ 4 becomes 7 g e q xg e q − 2 .
Express the solution in interval notation: [ − 2 , 7 ] .
Explanation
Problem Analysis We are given the inequality 2∣5 − 2 x ∣ − 3 ≤ 15 and asked to find the solution set.
Isolating the Absolute Value First, we isolate the absolute value term by adding 3 to both sides of the inequality: 2∣5 − 2 x ∣ − 3 + 3 ≤ 15 + 3 2∣5 − 2 x ∣ ≤ 18
Simplifying the Inequality Next, we divide both sides by 2: 2 2∣5 − 2 x ∣ ≤ 2 18 ∣5 − 2 x ∣ ≤ 9
Rewriting as a Compound Inequality Now, we rewrite the absolute value inequality as a compound inequality: − 9 ≤ 5 − 2 x ≤ 9
Subtracting 5 Subtract 5 from all parts of the inequality: − 9 − 5 ≤ 5 − 2 x − 5 ≤ 9 − 5 − 14 ≤ − 2 x ≤ 4
Dividing by -2 Divide all parts of the inequality by -2. Remember to reverse the inequality signs: − 2 − 14 g e q − 2 − 2 x g e q − 2 4 7 g e q xg e q − 2
Interval Notation Rewrite the solution in interval notation: − 2 ≤ x ≤ 7 x [ − 2 , 7 ]
Final Answer The solution to the inequality 2∣5 − 2 x ∣ − 3 ≤ 15 is [ − 2 , 7 ] .
Examples
Absolute value inequalities can be used in manufacturing to ensure that parts meet certain specifications. For example, a machine shop might need to produce bolts with a diameter within a certain tolerance of 10mm. This could be expressed as an absolute value inequality, where the actual diameter must be within a certain range of the target diameter. Similarly, in finance, absolute value inequalities can be used to model risk. For example, an investor might want to limit their portfolio's volatility to a certain level, which can be expressed as an absolute value inequality.
The solution to the inequality 2∣5 − 2 x ∣ − 3 ≤ 15 is [ − 2 , 7 ] . Therefore, the correct answer is option C. This solution was found by isolating the absolute value and using properties of inequalities to derive the final interval.
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